1
This is a preprint of a paper to appear in British Journal for the Philosophy of Science.
Everettian Confirmation and
Sleeping Beauty
Alastair Wilson
University of Birmingham & Monash University
email: [email protected]
ABSTRACT
Darren Bradley has recently appealed to observation selection effects to
argue that conditionalization presents no special problem for Everettian
quantum mechnics, and to defend the ‘halfer’ answer to the puzzle of
Sleeping Beauty. I assess Bradley’s arguments and conclude that while
he is right about confirmation in Everettian quantum mechanics, he is
wrong about Sleeping Beauty. This result is doubly good news for
Everettians: they can endorse Bayesian confirmation theory without
qualification, but they are not thereby compelled to adopt the unpopular
‘halfer’ answer in Sleeping Beauty. These considerations suggest that
objective chance is playing an important and under-appreciated role in
Sleeping Beauty.
1. Introduction
2. Confirmation in EQM
3. Sleeping Beauty
4. The selection model
5. Bradley’s argument
6. The right route to 1/3
7. The breakdown of the analogy
8. Alternative diagnoses
9. God’s Gambling Game
10. Non-chancy Sleeping Beauty cases
11. Conclusion
1. Introduction
In a recent BJPS paper (Bradley [2011]), Darren Bradley fans the flames of
the debate over confirmation theory in Everettian quantum mechanics (EQM).
Bradley argues that, by taking the centred nature of our evidence and the
corresponding observation selection effects into account in the right way,
Everettians can tell a plausible story about how EQM gets confirmed. The
spectre of automatic confirmation of many-worlds theories by any evidence
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whatsoever, which has worried authors like Barrett, Myrvold and Greaves, is
then dispelled without fuss; the ‘evidential problem’ for EQM is shown not to
need a solution of the sort proposed by Greaves [2007a] and by Greaves and
Myrvold [2010], which involves a framework of ‘quasi-credences’ and an
associated update rule called ‘quasi-conditionalization’. Even if quasi-credence
and quasi-conditionalization might be required for us to make sense of the pre-
measurement credential state of a subject who countenances EQM1, such notions
are unnecessary for modelling the experimental support that past measurement
results provide for the theory. Or so Bradley argues.
This discussion of centred evidence and observation selection effects in the
Everettian case, if correct, is significant enough by itself. But Bradley follows
Peter Lewis2 in drawing a close parallel between confirmation in EQM and the
puzzle of Sleeping Beauty (SB). According to Bradley, Lewis is right to think
that Everettians have to be ‘halfers’ when it comes to SB, but he is wrong to
think that this presents a problem for Everettians. Bradley in fact believes he
can demonstrate that halfing is the right response to SB. These are highly
controversial conclusions; how does Bradley reach them? In brief, his claim is
that in each case the strongest new evidence available to the relevant subject has
the form I learned that X by a random method. Given this conception of the
evidence, Bradley argues that EQM is not automatically confirmed and that
halfing is the correct response to SB. The key thesis here is that the procedure of
observation selection involved is in each problem a random one, in a sense to be
explained below. In defence of this thesis, Bradley offers an argument which
turns on the notion of a biased observation selection procedure.
In what follows I will argue that Bradley is right about EQM but wrong
about SB. Proper attention to observation selection effects does rule out any
automatic confirmation of EQM, but it does not generate the halfer position in
SB. The cases are disanalogous because Everettian confirmation scenarios
involve no equivalent of the chancy coin toss that governs Sleeping Beauty’s
awakenings. This is doubly good news for Everettians; their theory can be
confirmed by evidence in the usual way, but they need not be saddled with the
unpopular halfer conclusion.
In §2 and §3 I set out the details of two cases: an Everettian confirmation
scenario that I call ‘Quantum Wombat’, and the slightly modified Sleeping
Beauty scenario that Bradley calls ‘Technicolor Beauty’. §4 explains how
1 Wilson [forthcoming b] argues that they are not required even for this purpose.
2 See Lewis [2007] for details. Papineau and Durà-Vilà [2009] reply to Lewis; further
epicycles are Lewis [2009] and Papineau and Durà-Vilà [2009]. I discuss this exchange
in Section 8; my own account of this disanalogy between Everettian confirmation
and Sleeping Beauty is given in Section 7.
3
Bradley models these cases as selections from a population, and §5 reconstructs
his argument, which uses the selection model to support halfer-friendly and
Everettian-friendly conclusions. In §6, I explain how I think thirders should
respond. §7 is the core of the paper: in it I argue that the analogy between SB
and confirmation in EQM breaks down in a crucial way. In §8 I compare my
account of the disanalogy to extant accounts, and in §9 I describe a bizarre
variably-many-worlds theory which genuinely is analogous to SB. §10 applies my
analysis to a variant on SB which lacks the chancy element of the original case;
§11 is a conclusion.
2. Confirmation in EQM
Confirmation theory in EQM is perplexing because the theories to be
compared – EQM and some candidate one-world stochastic theory ST – have
systematically different consequences for the number of observers in existence.
Consider the following case:
Quantum Wombat: Wombat is unsure whether EQM or ST is
correct. He has just performed a spin measurement with
possible outcomes Up and Down, but he has not yet looked at
the result. According to EQM, after the measurement there are
two observers, located on branches of equal weight3, one of
whom observes Up and the other of whom observes Down.
According to ST, after the measurement there is one observer,
who observes either Up or Down, with equal probability.
Wombat is not sure whether i) EQM is true and he is one of
the two observers, or ii) ST is true and he is the only observer.
3 The metaphysics and epistemology of branch weight are controversial: see part 3
of Saunders et al. [2011] and Wilson [forthcoming b] for discussion. However, for our
purposes only the relatively straightforward equally-weighted case will be needed.
4
Suppose Wombat observes Up. Should he take this observation to support
EQM over ST? Plausibly not: the two theories are usually supposed to be (at
least approximately) empirically equivalent. However, the probability that Up is
observed by somebody is 1 according to EQM but is less than 1 according to ST.
Accordingly, the credential update rule which Greaves [2007a] and Bradley
[2011] dub ‘naïve conditionalization’ appears to break down when one of the
options on the table is EQM. The following line of thought is tempting: if the
many-worlds theory predicts all possible outcomes, then no possible observation
can disconfirm EQM; and since no one-world theory likewise predicts all possible
outcomes4, every observation confirms EQM over ST. In Bradley’s words:
The Ancients could have worked out that they have overwhelming
evidence for MWI merely by realizing it was a logical possibility and
observing the weather.
Bradley [2011], p.336
Something has gone wrong. And notice that it wasn’t any assumption that
EQM was correct that landed us in this mess; it was just the assumption that
EQM might be correct. Compensating by setting prior credences in EQM lower
is an unattractively brute-force response, since branching is so ubiquitous5. We
can only safely ignore the apparent problem that EQM generates for
confirmation theory if we are willing to set our prior credence in EQM arbitrarily
low, effectively ruling out EQM a priori. This seems an uncomfortably dogmatic
position for philosophers to adopt, given how seriously physicists take EQM.
3. Sleeping Beauty
Confirmation in EQM is analogous in certain ways to the Sleeping Beauty
problem, introduced to philosophers by Elga [2000]. I will describe the variant6
that Bradley calls ‘Technicolour Beauty’. Here is the setup.
4 I leave aside what Bostrom [2001] calls ‘big-world cosmologies’: single-world
theories in the traditional sense, but where a spatial or temporal infinity ensures that
some plenitude of qualitative possibilities is realized.
5 The issue of how, if at all, to quantify branching in EQM is a vexed one: see
Wallace [2012] and Greaves [2007b]. However, given that decoherence takes hold on
timescales in the order of 10-20 seconds (Zurek [2003] gives some accessible models),
any reasonable criterion will require initial prior credence in EQM to be well below
10-20 in order to insulate us from the automatic confirmation effect.
6 During the interval between waking and seeing the paper, Technicolour Beauty
reproduces the original Sleeping Beauty scenario. Accordingly, a full solution to
Technicolour Beauty determines a solution to the original Sleeping Beauty scenario,
while a solution to the original Sleeping Beauty scenario constrains (though perhaps
not uniquely) a solution to Technicolour Beauty. From this point on I will
concentrate on Technicolour Beauty, and will refer to it as SB for short. I hope that
the consequences of my argument for the original Sleeping Beauty scenario are clear.
5
Technicolour Beauty: Beauty will be put to sleep on Sunday
night and a fair coin tossed. If the coin comes up Heads, Beauty
will be woken on Monday. If the coin comes up Tails, Beauty
will be woken on Monday and on Tuesday. Beauty’s memory of
her Monday experience will be erased on Monday night; so each
waking is initially subjectively indistinguishable from every
other. However, shortly after each waking Beauty will be shown
either a Red or a Blue piece of paper. If Tails comes up, she will
be shown Red on Monday and Blue on Tuesday; if Heads comes
up, she will be shown either Red or Blue on Monday, depending
on the toss of a further fair coin. Beauty knows all this. Beauty
sleeps. Beauty wakes. Beauty is shown the paper. It is Red.
What should her credence be that the coin landed Heads, before
and after seeing that the paper is Red?
Confirmation scenarios in EQM have a structural similarity to Technicolour
Beauty. Let a spin measurement be made, which according to EQM results in an
Up branch and a Down branch and according to ST results in a stochastic
transition to either Up or Down. Then EQM predicts two branches containing
Up and Down, just as Tails predicts two days containing Red and Blue; ST
predicts one branch containing Up or Down, just as Heads predicts one day
containing Red or Blue. Bradley provides the following table illustrating the
analogy (he uses the acronym MWI to refer to EQM):
Bradley [2011] p. 333
In Section 7 I will argue that the analogy between SB and confirmation in
EQM breaks down; but I will pretend that it holds for the time being, while I set
out Bradley’s central argument.
4. The selection model
Bradley models the procedure of making an observation as the taking of a
sample from a population of locations of observation. A location of observation is
an agent’s having a certain perceptual experience at a particular spacetime
location (and, if EQM is correct, on a particular branch).
If ST is correct, then when Wombat observes the result of the spin
measurement the population to be sampled contains one location of observation,
which is either an Up location or a Down location. If EQM is correct, then when
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Wombat observes the result of the spin measurement the population to be
sampled contains two locations of observation, one of which is an Up location
and the other of which is a Down location. Wombat’s discovery of the result is
then represented as him sampling himself from a population of two wombat-
stages, one in each branch.
If the coin lands Heads, then the population to be sampled when Beauty
observes the colour of the paper contains one location of observation, which is
either a Red location or a Blue location. If the coin lands Tails, the population
to be sampled when Beauty observes the colour of the paper contains two
locations of observation, one of which is a Red location and the other of which is
a Blue location. Beauty’s discovery of the colour is then represented as her
sampling herself from a population of two person-stages, one on Monday and one
on Tuesday.
Note that there is no dubious generalized indifference principle being used
here: it is not assumed that each epistemic possibility must be assigned the same
credence. The route to 1/3 in Sleeping Beauty which starts from the observation
that there are three epistemically possible experiences and argues they must be
equiprobable is not endorsed by any thirders I know of; and if anything, this
argument is even less convincing in the case of Everettian confirmation. The
population from which the observations are sampled is, in both examples, a
population of experiences which do in fact occur: in different branches in one
case and on different days in the other.
With this caveat, Bradley’s sampling model looks innocuous. In sampling a
single member from a population, one knows that the sample will be a member
of the population, but one doesn’t know which member it will be. If one does not
know which population the sample is being drawn from, this creates further
uncertainty. Prior to making their observations, Beauty and Wombat do not
know which population their observation is to be drawn from, and they do not
know which member will be drawn if the population contains two members. This
is all there is to the sampling model, and I can’t see any grounds for disputing
its applicability.
Why does the sampling model help? Bradley’s answer is that it highlights
the role of the observation selection effect produced by the nature of the
observation selection procedure that selects a sample from the population. We
often know not only what the sample is – what the content of our location of
observation is – but also how we acquired it – how the location was picked out
from the population. An observation selection procedure, or a method for short,
is a particular way of drawing a sample from a population of locations of
observation. Bradley argues that when we make an observation of X, the total
evidence we acquire is not just X occurred, and not even just I learned that X
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occurred, but I learned that X occurred by method M. This of course entails that
X occurred, but it can entail more or less in addition. For example, if method M
only selects very small objects, then learning that it has selected a very small
object is less revealing about the population than it would be if method M could
select an object of any size. Insofar as knowledge of the method used affects the
inferences that can be drawn after making a particular observation, we say that
that method produces an observation selection effect.
Let’s apply this to our examples. On Bradley’s view, the evidence that
Wombat gains when he sees either Down or Up has the form I learned that X
either by ST being true and X being the outcome produced by a stochastic
process or by EQM being true and X being an outcome seen on one of the
branches. The evidence that Beauty gains when she sees either red or blue paper
has the form I learned that X either by Heads landing and X being seen on the
one awakening, or by Tails landing and X being seen on one of the two
awakenings. In both cases this evidence is stronger than just X; and as
Bayesians, we know always to conditionalize on the strongest evidence available7.
Not every method of sampling from a population is a random method.
Bradley cites Eddington’s famous example of the net which can only catch fishes
larger than a certain size, and adds the example of an urn of balls with an
opening through which only some of the balls can fit. Bradley contends that in
the cases – Tails and EQM – where the populations sampled contain more than
one member, we should treat the samplings as random. Then Beauty should be
equally confident that it is Monday as that it is Tuesday conditional on the coin
landing Tails, and Wombat should be equally confident that he is on an Up
branch as that he is on a Down branch conditional on EQM being correct.
What is the probability of the new evidence acquired via our agents’
observations, given the various hypotheses being tested? Bradley argues that in
the quantum case the probability of seeing Up was 1/2, whether ST or EQM is
true (likewise for Down); and that in the SB case the probability of seeing Red
was 1/2, whether the coin landed Heads or Tails (likewise for Blue). Then in
both cases the new evidence has the same probability on either hypothesis, so
neither hypothesis is confirmed over the other in either case. Observing the
colour of the paper does not provide Beauty with support for Tails; and
observing the result of a spin measurement does not provide Wombat with
support for EQM.
7 Failure to conditionalize on an agent’s strongest evidence is a failure of rationality
for Bayesians: it leads to fallacies of exclusion, as when someone argues from their
unblemished safety record on the road to the conclusion that it would be safe to
drive home, neglecting the eight pints of beer recently consumed.
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The claim that the probability of whatever new evidence is acquired is 1/2
on both of the hypotheses to be tested is the crux of Bradley’s argument. Once it
is established, automatic confirmation of EQM drops out of the picture, and the
argument that Bradley discusses for the thirder solution lapses. So we must
examine the motivation for this key claim. I’ll initially consider only the case of
Everettian confirmation; once I’ve ironed out some wrinkles in the argument, I’ll
move to consider SB also.
5. Bradley’s argument
Bradley defines a few technical terms for use in his argument. An
observation selection procedure is biased towards X iff: if X is in the population,
then X is selected for the sample. (He notes that this might be more aptly called
‘maximal bias’.) An observation selection procedure is random if each item in the
population has an equal chance of being selected for the sample.
Armed with this terminology, Bradley proceeds as follows.
1. If EQM is correct, the observation made by Wombat can be modelled
as a sampling from a population of two locations of observation – the
Up branch and the Down branch. (Premise.)
2. If EQM is correct, after the measurement Wombat is only located on
one branch, so he only observes a single outcome; he observes either
Up or Down but not both. (Premise.)
3. If the observation selection procedure which selects Wombat’s
observation is biased towards both Up and Down, then both Up and
Down are selected and Wombat observes both Up and Down. (From 1,
definition of biased towards X.)
4. If EQM is correct, the observation selection procedure which selects
Wombat’s observation cannot be biased towards both Up and Down.
(From 2, 3.)
5. If a selection procedure is not biased towards both of its possible
outcomes, then it is random. (?)
6. If EQM is correct, the observation selection procedure which selects
Wombat’s observation is random. (From 4, 5.)
7. If ST is correct, the observation selection procedure which selects
Wombat’s observation is random. (Premise.)
8. Wombat knows that either ST or EQM is correct. (Premise.)
9. The strongest new evidence acquired by an agent on making an
observation is ‘I learned that X by method M’. (Premise.)
10. The strongest new evidence EW acquired by Wombat on making his
observation is ‘I learned that (Up or Down) by a random procedure. ’
(From 6, 7, 8, 9.)
11. For a random procedure, the probability of any possible outcome is
equal. (Definition of random procedure.)
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12. The probability of Wombat’s strongest new evidence given EQM is
1/2. (From 10, 11.)
13. The probability of Wombat’s strongest new evidence given ST is 1/2.
(From 10, 11.)
14. Observing the result of the measurement does not confirm EQM over
ST. (From 12, 13.)
Wombat’s observation – whatever the result – does not confirm EQM over
ST, or vice versa. This is the intuitively correct result.
There is an obvious gap in the argument as stated, and it is rather
surprising that Bradley fails to plug it. In an introductory section he explicitly
notes that biased procedures and random procedures do not exhaust the possible
types of procedure; yet he moves freely from the conclusion that the observation
selection procedure which selects Wombat’s observation is not biased in both
directions, to the conclusion that it is random8:
Suppose instead that there is no bias. That is, we have a random
procedure - Up branches have the same probability of being observed
as Down branches, so you are just as likely to be in an Up branch as a
Down branch.
Bradley [2011], p. 331
For all that has been established so far, the procedure which selects
Wombat’s observation could be biased towards Up and not towards Down, or be
biased towards Down and not towards Up, or be biased towards neither outcome
but be such that Up is selected with probability 0.99. (Recall that ‘bias’, for
Bradley, means maximal bias: if X is in the population then X is definitely in the
sample.) So there seems to be no good motivation for 5, the premise that if a
selection procedure is not biased in both directions then it is random. Bradley
moves from lack of maximal bias in both directions (a trivial claim) to
equiprobability (a highly non-trivial claim).
Bradley apparently intends9 this gap in the argument to be filled by the
restricted principle of indifference proposed by Elga [2000]. This principle, which
Bradley devotes the final section of his paper to defending, is that two centred
8 Bradley [2007] uses ‘random’ to mean not-maximally-biased. The argument given
above does establish that the procedure is random in this weaker sense. But it
doesn’t establish that the procedure is random in the sense specified in the quote in
the main text; and it is this stronger conclusion which is needed to establish 12 and
13. 9 Bradley does say that he doesn’t want to rely too much on this principle, because
the thesis that the selection procedures in question are not maximally biased in
either direction doesn’t require it. However, the (much stronger) thesis that the
selection procedures are random does appear to require the indifference principle.
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worlds deserve equal credence when they correspond to the same uncentred
world and the agents at the centres are ‘subjectively indistinguishable’. However,
this particular indifference principle is a poisoned chalice for Everettians since it
conflicts with the Born rule, which says that agents should apportion their
credences to the branch weights. Probabilities must be given by the Born rule if
the predictions of EQM are to agree with the predictions of conventional
quantum theory; and so Everettians have made sustained attempts to argue for
it (see e.g. Deutsch [1999], Saunders [2004], Wallace [2002, 2006, 2010, 2012],
Wilson [forthcoming b]). Bradley stipulates that his arguments should only be
taken as applying to the case where the two branches have equal weight, thereby
avoiding clashing with the Born rule; but prima facie Elga’s restricted principle
of indifference applies to cases of unequal weight as well as to cases of equal
weight.
I think the best way to patch up Bradley’s argument in an Everettian-
friendly fashion is to appeal to the symmetry of the setup. In the case of equal
weight, the physical situation is exactly symmetric between the Up outcome and
the Down outcome. There are two possible outcomes; exactly one of them will be
observed by Wombat; there is no fact (such as an unequal weighting of the
branches) which could provide a reason why one outcome and not the other is
observed by Wombat; so the selection procedure by which Wombat’s
observation is selected must be random, on pain of arbitrariness. Bradley does
hint at this line of reasoning:
Up branches are just as hospitable to life, and just as likely to be
observed as Down branches.
Bradley [2011], p. 331
Inserting this sort of appeal to symmetry into Bradley’s argument gives rise
to the following modified argument, which can safely be endorsed by
Everettians:
15. If EQM is correct, the observation made by Wombat can be modelled
as a sampling from a population of two locations of observation – the
Up branch and the Down branch. (Premise.)
16. If EQM is correct, after the measurement Wombat is only located on
one branch, so he only observes a single outcome; he observes either
Up or Down but not both. (Premise.)
17. If the observation selection procedure which selects Wombat’s
observation is biased towards both Up and Down, then both Up and
Down are selected and Wombat observes both Up and Down. (From
15, definition of biased towards X.)
18. If EQM is correct, the observation selection procedure which selects
Wombat’s observation cannot be biased towards both Up and Down.
(From 16, 17.)
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19. If a selection procedure is not biased towards both of its possible
outcomes, and it is symmetric with respect to its possible outcomes,
then it is random. (Premise.)
20. If EQM is correct, the observation selection procedure which selects
Wombat’s observation is symmetric with respect to its possible
outcomes. (Premise.)
21. If EQM is correct, the observation selection procedure which selects
Wombat’s observation is random. (From 19, 20.)
22. If ST is correct, the observation selection procedure which selects
Wombat’s observation is random. (Premise.)
23. Wombat knows that either ST or EQM is correct. (Premise.)
24. The strongest new evidence acquired by an agent on making an
observation is ‘I learned that X by method M’. (Premise.)
25. The strongest new evidence EW acquired by Wombat on making his
observation is ‘I learned that (Up or Down) by a random procedure.’
(From 21, 22, 23, 24.)
26. For a random procedure, the probability of any possible outcome is
equal. (Definition of random procedure.)
27. The probability of Wombat’s strongest new evidence given EQM is
1/2. (From 25, 26.)
28. The probability of Wombat’’s strongest new evidence given ST is 1/2.
(From 25, 26.)
29. Observing the result of the measurement does not confirm EQM over
ST. (From 27, 28.)
The same argument can be applied to SB.
30. If the coin lands Tails, the observation made by Beauty can be
modelled as a sampling from a population of two locations of
observation: the Red day and the Blue day. (Premise.)
31. If the coin lands Tails, on awakening Beauty is only located on one
day, so she only observes a single outcome; she observes either Red or
Blue but not both. (Premise.)
32. If the observation selection procedure which selects Beauty’s
observation is biased towards both Red and Blue, then both Red and
Blue are selected and Beauty observes both Red and Blue. (From 30,
definition of biased towards X.)
33. If the coin lands Tails, the observation selection procedure which
selects Beauty’s observation cannot be biased towards both Red and
Blue. (From 31, 32.)
34. If a selection procedure is not biased towards both of its possible
outcomes, and it is symmetric with respect to its possible outcomes,
then it is random. (Premise.)
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35. If the coin lands Tails, the observation selection procedure which
selects Beauty’s observation is symmetric with respect to its possible
outcomes. (Premise)
36. If the coin lands Tails, the observation selection procedure which
selects Beauty’s observation is random. (From 34, 35.)
37. If the coin lands Heads, the observation selection procedure which
selects Beauty’s observation is random. (Premise)
38. Beauty knows that the coin lands either Heads or Tails. (Premise)
39. The strongest new evidence acquired by an agent on making an
observation is ‘I learned that X by method M’. (Premise)
40. The strongest new evidence EB acquired by Beauty on making her
observation is ‘I learned that (Red or Blue) by a random procedure. ’
(From 36, 37, 38, 39.)
41. For a random procedure, the probability of any possible outcome is
equal. (Definition of random procedure.)
42. The probability of Beauty’s strongest new evidence given Heads is 1/2.
(From 40, 41.)
43. The probability of Beauty’s strongest new evidence given Tails is 1/2.
(From 40, 41.)
44. Observing the colour of the paper does not confirm Tails over Heads.
(From 42, 43.)
How should thirders respond to this argument? I think they should grant
the conclusion, but deny that it undermines their position. Bradley’s argument is
effective against someone who maintains that Beauty should have an initial
credence in Heads of 1/2, but who also maintains that this credence should alter
to 1/3 as soon as she sees any evidence which she is not guaranteed to see (such
as the red or blue paper, or cloud patterns outside her window). However, this
would be an unattractive position for a thirder to adopt. Sensible thirders ought
to say that Beauty’s initial credence on waking, before she opens her eyes and
sees anything, should already be 1/3, and that subsequent evidence should not
automatically alter that credence.
What Bradley’s argument does achieve, in the case of SB, is to show that
thirders cannot motivate their position by arguing that some new evidence that
Beauty acquires after awakening supports Tails over Heads. So Bradley’s
argument does undermine (Bradley’s reconstruction of) the view of Titelbaum
[2008], which motivates the answer 1/3 via the thought that Red is guaranteed
to be seen if the coin lands Tails. But thirders can, and should, motivate their
solution differently.
6. The right route to 1/3
In this section I will say how I think the 1/3 answer should be motivated
for the case of Technicolour Beauty. This solution will also entail that the 1/3
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answer is correct in the original Sleeping Beauty scenario. The resulting position
is essentially that of Elga [2000], although I put a slightly different gloss on it.
Thirders should agree that EB is the strongest new evidence that Beauty
acquires on making her observation of the coloured paper. And they should agree
that the probability of EB given Heads is the same as the probability of EB given
Tails, and hence that Beauty’s observation of the colour of the paper does not
provide her with support for Tails. But they should deny that the total evidence
that Beauty has after observing the colour of the paper is her evidence on
Sunday night plus EB. For Beauty’s evidential state has undergone an additional
change overnight. She has lost centred evidence, evidence about which day it is.
As a result of this evidence loss, she may no longer treat the result of the coin
toss as effectively chancy, in a sense to be explained below10.
Both my claim that Beauty has lost evidence and my claim that this has
consequences for her credences about the outcome of the coin toss are
contentious. In this section I will sketch the basis for thinking that Beauty has
lost evidence, and then explore the consequences of this claim for Beauty’s
credential state.
Take first the claim that Beauty has lost irreducibly centred evidence
overnight on Monday. This evidence loss can be most straightforwardly
expressed by saying that she has lost evidence about which day it is now; it
cannot be expressed in any fragment of language which does not contain
indexical terms. Before going to sleep Beauty knew what day it was then – it
was Sunday. On awakening she does not know which day it is now – it could be
Monday or Tuesday. Normally, as the days pass without our losing track of their
passing, we lose some centred evidence but gain corresponding new centred
evidence. In cases like Sleeping Beauty, or like the case of O’Leary described by
Elga [2004], there is a loss of irreducibly centred evidence which is not
adequately compensated for. See Dieks [2007] for further discussion of this point.
Of course, to point to a way in which Beauty’s evidential state changes
overnight is not yet to show that she should have credence 1/3 in Heads on
waking. The change in evidential state might simply be irrelevant, or it might be
relevant in a way which results in a credence other than 1/3 in Heads. This
brings us to the second claim; that the overnight change in evidential state
means that Beauty should no longer regard the coin toss as effectively chancy.
To explain what I mean by this, it will help to start by rehearsing one of the
arguments given in Elga [2000] for the answer 1/3.
10 See p.16 for the precise definition of this term.
14
At the heart of this argument are three assumptions11. The first assumption
is that chance is the norm of credence: that in a situation where an agent knows
the chances and has no inadmissible information12, the agent’s credences should
match the chances. Where a fair coin toss is in the future, an agent cannot have
inadmissible information about it without the help of precognition or some other
form of backwards causation.
The second assumption is that it doesn’t matter when the coin is tossed, as
long as it has been tossed by Tuesday morning. In the original version of SB, the
coin is tossed on Sunday night. But since Beauty will be woken on Monday
regardless, and the procedural need for the information about the result of the
coin does not arise until the experimenters are deciding whether to wake Beauty
on Tuesday morning, the coin could equally well be tossed on Monday night.
Beauty knows this, so she ought to adopt the same credence on waking if she
knows the coin is tossed on Monday as she adopts if she knows the coin is tossed
on Sunday. (Indeed, it should make no difference to her credences if the coin is
tossed even earlier, say on Saturday night.)
The third assumption, which Elga motivates via his restricted principle of
indifference, is that the probability of it being Monday conditional on Tails is
the same as the probability of it being Tuesday conditional on Tails. Whatever
we may think about the restricted principle of indifference in general (and I have
suggested above that it would be a mistake for Everettians to adopt it
unrestrictedly), this particular application of it seems unproblematic. Beauty
cannot distinguish between a Monday waking given Tails, and a Tuesday waking
given Tails. She thus has no reason to assign them different credences, and so
her credence that it is Monday conditional on Tails should be the same as her
credence that it is Tuesday conditional on Tails.
Together, these assumptions entail that the thirder solution is correct. In
the case in which the coin is tossed on Monday night, the chance on Monday of
the coin landing Heads is 1/2. Beauty knows this, so (by the first assumption)
on awakening her credence in the coin landing Heads conditional on it being
Monday is 1/2. On awakening, Beauty does not know whether a) it is Monday
(in which case the toss is in the future, and the chance of Heads and of Tails are
both 1/2) or b) it is Tuesday (in which case the toss has already occurred, and
the chance of Heads is 0 and the chance of Tails is 1).
11 One further assumption, denied inter alia by Lewis [2010], is that on awakening
Beauty’s credence that the coin lands Heads cannot come apart from her credence
that this is a Heads-waking. The motivation for this assumption ought to be clear:
the setup of the case ensures that on awakening Beauty knows that these two
propositions have the same truth-value, so she should assign them equal credence.
12 See Lewis [1980] and Hoefer [2007] for classic discussions of the notion of
inadmissible information.
15
Using a standard notion for subjective credence: Cr(Heads|Monday) = 1/2,
by the Principal Principle. Cr(Heads|Tuesday) = 0, since Beauty is awakened on
Tuesday only if the coin lands Tails. So if Cr(Tuesday) > 0, then Cr(Heads) <
1/2. This is already enough to establish the falsity of the halfer conclusion. By
the third assumption and an application of Bayes’ rule, the thirder conclusion
follows.
The above argument applied to the case in which the coin is tossed on
Monday night. But by the second assumption, the thirder conclusion is correct
for the case where the coin is tossed on Monday night if and only if it is correct
for the case where the coin is tossed on Sunday night. So the thirder conclusion
follows for these cases also.13
On awakening, Beauty’s uncentred evidence does not change, but her
centred evidence does change: Elga’s argument shows how this change results in
a shift in credence in Heads from 1/2 to 1/3. The remaining difficulty is that of
explaining why this evidential change should produce such a credential shift.
After all, we are constantly undergoing changes in centred evidence which have
no significant effect on our credences in uncentred propositions.
I suggest that the difference is that in SB the overnight loss of centred
evidence is hooked up to a chancy memory-loss process. This means that
different inferences can be drawn from the available self-locating evidence before
and after going to sleep on Sunday. To explain this, I think we can usefully
appeal to the notion of a proposition’s being effectively chancy. On Sunday
night, it is effectively chancy for Beauty whether Heads; on awakening, it is no
longer effectively chancy whether Heads.
Effective chanciness isn’t the same as chanciness in the sense of Lewis
[1980], according to which all past propositions have chance zero and the chance
of a proposition cannot vary from person to person. For a proposition to be
effectively chancy is both a person-dependent and a time-dependent matter. The
same proposition can be effectively chancy for one person and not for another (as
when the latter but not the former has observed the result of a coin toss), and it
can be effectively chancy for a given person at one time and another (before and
13 Arntzenius [2003] resists this argument on the grounds that Beauty knows she
may suffer a ‘cognitive mishap’ on Monday night, and hence that she will not
necessarily remain an ideal Bayesian agent. But it is unclear why this means that she
should not do the best she can, by conditionalizing on her strongest evidence
whenever possible. Arntzenius also objects that the argument hooks up Beauty’s credence in Heads to the issue of whether she accepts causal or evidential decision
theory. I do not find this connection either surprising or worrisome. (Nor does Briggs
[2010].)
16
after observing the result of the coin toss, respectively). As I use the phrase, for
a proposition P to be effectively chancy for agent A at time t is for P to be or
have been chancy, and for A to possess no evidence at t which is or might be
inadmissible with respect to A.
Since effective chanciness is an agent-dependent and time-dependent
business, evidence that some proposition is effectively chancy is irreducibly-
centred evidence. It is evidence that the proposition is effectively chancy now,
for me. Such evidence will not be treatable in a framework where all evidence
takes the form of untensed propositions with timeless truth-values. This is no
great surprise: in response to SB, several authors have attempted to develop
more general accounts of credential updating which can cope adequately with
centred evidence. Such accounts can be found in Titelbaum [2008], Meacham
[2008], Bradley [2011], Schulz [2010] and Schwarz [2012].
I assume that if a proposition P is effectively chancy for A at t, then at t A
is bound indirectly by the Principal Principle with respect to T. Even though P
may no longer be chancy at t, if P was chancy at some earlier time and if A
both knows the chance that P had and has no evidence inadmissible with respect
to P, then at t A is still bound to match her credence to the value that the
chance of P is known to have had.
The notion of effective chanciness gives us the resources to say what sort of
change in Beauty’s evidence occurs overnight on Sunday. Where on Sunday
night the result of the coin toss is effectively chancy for Beauty, when she
awakes on Monday it is no longer effectively chancy. For all she knows on
awakening, it could now be Tuesday, in which case the coin would have to have
already been tossed and landed Tails. In that case, Beauty’s new centered
evidence on awaking is inadmissible with respect to the coin toss. So Beauty has
lost evidence that the result of the coin toss is an effectively chancy matter,
because she has lost evidence that she has no inadmissible evidence. As a result,
she is no longer constrained by the Principal Principle to match her credences to
the objective chances.
Since the only scenario in which Beauty possesses any inadmissible evidence
is a scenario in which the coin landed Tails and it is Tuesday, learning that the
coin toss is no longer effectively chancy should alter her credences in the
direction of Tails. By the second assumption of the argument given above and
an application of Bayes’ rule, she should end up with credence 1/3 in Heads.
Armed with this result, we can consider what would happen if Beauty
learns that it is Monday. When she does, she learns that she has no evidence
that is inadmissible with respect to the coin toss. She thereby re-acquires the
evidence she lost overnight: that the coin toss is effectively chancy. Accordingly,
she is once again bound by the Principal Principle with respect to the chance of
17
Heads, and must adopt credence 1/2 in Heads. Although this line of thought has
been developed in the context of Technicolour Beauty, it is essentially the same
as Elga’s solution to the original Sleeping Beauty case.
One way out for halfers is to maintain that it really matters whether the
toss is past or future. Halfers are committed to the idea that if Beauty learns
that it is Monday, she should increase her credence to 2/3 that the coin will land
Heads. Since it is crazy to think of a future fair coin toss that it is 2/3 likely to
land Heads14, halfers must say there is a difference between the toss-on-Sunday
and toss-on-Monday cases. The resulting view has it that in a case in which
Beauty knows the coin is tossed Sunday night, halfing is the correct solution; but
that in a case in which Beauty knows the coin is tossed Monday night, thirding
is the correct solution. This requires Beauty to bet differently depending on
whether a coin toss is future or past, even when she knows that the toss is fair15.
I will not investigate here how halfers might seek to sweeten this pill.
In this section I have shown how thirders can motivate their position while
simultaneously accepting Bradley’s conclusions about the nature of the new
evidence Beauty acquires on seeing the coloured paper, and about the bearing of
this evidence on the result of the coin. The next section argues that the
analogous move fails in the case of automatic confirmation in EQM. The analogy
between Sleeping Beauty and Quantum Wombat breaks down, since there is no
chanciness in the latter case to correspond to the chanciness of the coin toss.
7. The breakdown of the analogy
Beauty loses relevant centred evidence, and therefore should not continue to
treat the result of the coin toss as effectively chancy. However, Wombat loses no
similarly-relevant centred evidence. Wombat does not know, in advance of
performing the spin measurement, that whether ST or EQM is true will be fixed
by the result of a future fair coin. Indeed, Wombat knows that this is not the
case – whichever theory is true is already true, and has chance 1. As a result, it
is not the case that whether ST or EQM is true ceases to be an effectively
chancy matter on performing the spin measurement, and Wombat’s credences in
ST and EQM should not change. EQM is not automatically confirmed.
14 Crazy or not, this was the view defended by Lewis [2001]. According to Lewis, on
discovering that it is Monday Beauty acquires knowledge which is inadmissible with
respect to a future coin toss. Given that SB involves no precognition or any other
variety of backwards causation, this suggestion is implausible.
15 What if the time of the toss itself is unknown: for example, if it will occur during
the first 10 minutes of the Monday waking? What if it is unknown in a self-centred
way: if it will occur 10 minutes after the Monday waking, and there is no clock in
Beauty’s room? I leave these awkward questions to halfers.
18
The disanalogy between SB and QW is, on reflection, a straightforward one.
Whether EQM or ST is true does not depend on any chance process, and
Wombat knows that. In contrast, whether the coin lands Heads or Tails does
depend on a chance process, and Beauty knows that. Consequently Beauty loses
relevant evidence when she is put to sleep, but Wombat loses no relevant
evidence on performing the measurement. The effect which generates the answer
1/3 in the case of Sleeping Beauty is absent in the case of Everettian
confirmation scenarios.
Wombat is correct to think that the probability of whichever experimental
result he sees given ST is the same as the probability of his new centred evidence
given EQM. And Beauty is correct to think that the probability of whichever
colour of paper she sees is the same given Heads as it is given Tails. But
Wombat can, after the measurement, still take into account all the evidence he
had before the measurement that was relevant to EQM vs. ST. Beauty cannot,
after awakening, still take into account all the evidence she had before going to
sleep that was relevant to Heads vs. Tails. For some of that was irreducibly
centred evidence: that the result of the coin toss was effectively chancy.
Bradley at one point implies that Wombat does lose information, endorsing
Lewis’ evocative phrase ‘gets lost in the branches’ (Bradley [2011] p. 333). The
idea seems to be that before the spin measurement, Wombat knows exactly
where he is – on the initial branch, about to press the button, while afterwards
he could be in either of two places:
The problem with this response is that the information loss in question is
irrelevant. It doesn’t provide evidence which bears on whether a particular
proposition is currently effectively chancy, so it does not give rise to any thirder-
style shift in credence.
19
The irrelevance of the self-locating uncertainty which Lewis and Bradley
point to is underlined by considering the diverging interpretation of EQM,
defended by Saunders [2010] and by Wilson [forthcoming a]. According to
overlapping EQM, different branches have earlier segments in common: the
‘splitting worlds’ metaphor is apt. According to diverging EQM, different
branches are mereologically isolated: the ‘parallel worlds’ metaphor is apt.
Diverging worlds may match one another up to a time, but they have no
segments in common. Wombat, pre-measurement, is on one branch and one
branch only, although he has no idea whether it is a branch on which Up will be
measured or a branch on which Down will be measured. He is just as lost in the
branches as he is after the measurement. Pictorially:
The evidence which defenders of the analogy must suppose that Wombat
loses is evidence which, on the diverging picture, Wombat never even had. But
the choice between branching and diverging versions of EQM, even if it is
relevant to pre-measurement uncertainty16, should not be relevant to the bearing
of past observations on the likelihood of a many-worlds theory. It is a problem
with the analogy that it says that the choice between branching and divergence
is relevant in this way: the analogy entangles issues of metaphysics and
epistemology which are better kept apart.
I have argued that the source of the disanalogy between confirmation of
EQM and Technicolour Beauty is that whether Heads or Tails is true is
determined by a chance process, while whether EQM or ST is true is not so
determined. In the next section I compare this suggestion to some alternative
accounts of the disanalogy.
16 Wilson [forthcoming a] argues that it is.
20
8. Alternative diagnoses
Other authors to diagnose disanalogies between Everettian confirmation and
SB are Papineau & Durà-Vilà [2009a, 2009b] and Peterson [2009]. As far as I can
see, these authors give competing accounts of the disanalogy, both of which
differ from my own account. The issue is complicated because these authors
primarily discuss the ‘simplified Sleeping Beauty’ case presented in Lewis [2007],
which involves no coin toss but rather has Beauty awaken on both Monday and
Tuesday, with her memory erased in between. The simplified Sleeping Beauty
case is not analogous to Everettian confirmation scenarios, but is supposed to be
analogous to branching events for agents who are certain of the truth of EQM.
Papineau & Durà-Vilà take the difference between the simplified SB case
and the case of Everettian branching to be that there are ‘two branches of
reality after the spin measurement’, while there is ‘only one branch of reality in
the simplified Sleeping Beauty case’. They claim that this allows Everettians to
rationally assign credences of less than 1 to each outcome of the spin
measurement, while Beauty in the simplified scenario must assign credence 1 to
waking on both Monday and Tuesday. However, this claim appears to rest on a
conflation of the probability that someone wakes on Monday with the
probability that it is now Monday for a just-awakened agent. Papineau and
Durà-Vilà correctly claim that the probabilities that someone wakes on Monday
and that someone wakes on Tuesday are both 1 in the simplified Sleeping
Beauty case. But likewise, the probabilities that someone sees Up and that
someone sees Down are both 1 in the case of Everettian branching. Papineau &
Durà-Vilà have not explained why the difference between days and branches
leads to a difference between Everettian branching and the simplified Sleeping
Beauty scenario.
Once we resist the conflation of the hypotheses that it is now Monday and
that someone is awake on Monday, then (as Lewis [2009] replies) it is altogether
unclear why the metaphysical difference between days and branches to which
Papineau and Durà-Vilà point is a relevant difference. After the measurement
but before the results have been examined, Wombat knows that – if EQM is
correct - he is one of two observers, but does not know which. Likewise, after
awakening Beauty knows that – if the coin landed Tails – she is on one of two
days, but does not know which.
I suspect that the reason that Papineau and Durà-Vilà locate the disanalogy
where they do is that they have been distracted by the pre-measurement
credential state of the Everettian subject. On the account of Everettian
probability defended by Papineau (see Papineau [1996], [2010]), the pre-
measurement subject should set her credences according to the branch weights
even though she knows, with certainty, that there will be an Up branch and that
21
there will be a Down branch. But the analogy between SB and confirmation in
EQM can be made out entirely in terms of the post-measurement state of the
observer, so pre-measurement credences are beside the point.
Peterson [2011] offers an alternative diagnosis of the disanalogy. Although
his discussion is rather hedged, the suggestion appears to amount to this. On
awakening, Beauty is uncertain about whether she has been previously
awakened; however, on making a measurement, an agent certain that EQM is
correct is sure that there is another branch on which another agent observes
another result. Beauty’s two wakings are connected in some way - Peterson
suggests either by personal identity or by causal or counterfactual dependency -
in which two agents in different branches are not. Why is this connection
epistemically relevant? According to Peterson, it is because this connection
provides something additional for Beauty to be uncertain about - whether or not
she has been previously awakened. Unfortunately, Peterson doesn’t say why this
additional uncertainty should affect Beauty’s credences about the result of the
coin. So as things stand, his account of the source of the disanalogy is no
advance on that of Papineau and Durà-Vilà.
The diagnosis of the disanalogy that Peterson offers does have a distinctive
consequence, which points to a potential way of motivating this differential
treatment of times and worlds. Following through on his account of the
disanalogy, Peterson suggests that in SB variant cases where the agents who
awaken after a toss of tails are distinct subjects, rather than being distinct time-
slices of the same subject, the motivation for being a thirder disappears:
...in the Beauty case, the thirder’s position seems correct; however,
were the situation to involve not one person waking up twice but two
successors being awoken once each... the halfer solution would be the
correct one.
Peterson [2009]
A similar view is defended by Schwarz [MS], who argues that the correct answer
to SB depends on whether it is construed as an episode of fission or not - that is,
on whether the Tuesday waker should be regarded as a ‘successor’ of the
Monday waker. Schwarz motivates the view from general considerations about
diachronic belief updating17; and Peterson could help himself to this sort of
motivation.
However, the resulting picture is highly problematic. Personal identity over
time is a vague and arguably context-dependent business, and we can run a
17 Related arguments are given by Meacham [2010]. Meacham has confirmed (p.c.)
that he accepts Schwarz’ conclusions about rational belief in fission cases, though
he’d resist Schwarz’ further claims about chance and admissibility.
22
Sorites series from cases in which the sleeper who awakes on one day is definitely
continuous with the sleeper who awakes on the next day to cases in which they
are definitely distinct. Peterson and Schwarz are committed to saying that
somewhere along this series there is a sharp discontinuity, where the credence
that the subjects should have alters from 1/3 to 1/2, even though the subjects
may be unaware of where this discontinuity lies.
Moreover, this view does not altogether escape the pathological
consequences associated with the halfer solution. For example, Schwarz admits
that on his account one of two fission products, about to toss a fair coin to
determine whether the other fission product will be destroyed before awakening,
should have 2/3 confidence that the fair coin will land Heads (Schwarz [MS]
p.22). I take this to be a fatal difficulty for the approach; so I conclude that
Peterson has not correctly located the disanalogy between Everettian
confirmation and SB.
9. God’s Gambling Game
In Section 7 I argued that the analogy between SB and confirmation in
EQM breaks down because the result of the coin toss depends on a chancy
process, while whether EQM or ST is correct does not depend on any chancy
process. The analogy can be restored if we consider a modified branching theory
which I will call God’s Gambling Game:
God’s Gambling Game: Whenever a spin measurement is made, God
tosses a fair coin to select either Down or Up. He then creates a branch
in which this selected outcome is observed. He then tosses a further fair
coin. If it lands Heads he rests, his creative work done. If it lands Tails
he creates a second branch, in which the outcome not selected by the
first coin toss is observed.
This scenario is illustrated below:
23
Call the theory that God does carry out the procedure just outlined GGG.
Since this situation genuinely is analogous to the Technicolour Beauty scenario,
does an analogue of the thirder effect arise? The answer is yes. If an agent knows
that GGG is true, then after making a measurement, then even before he
observes the result, that agent should become uncertain as to whether he is on
the first branch to be created (in which case the second coin might have landed
Heads or Tails), or whether he is on the second branch (in which case the second
coin must have landed Tails.) On making the measurement, then, the agent loses
the evidence that whether there will be one branch or two is an effectively
chancy matter.
Does this lead to a problematic automatic confirmation for GGG? The
answer is no. It’s true that, via the standard thirder reasoning run through
above, evidence that experiments have taken place does raise the probability
that the world has multiplied conditional on GGG. But this result is irrelevant
to our credence in GGG itself. Reason to reassign credence amongst the various
ways things could go conditional on a theory isn’t reason to change
unconditional credence in that theory. This applies equally to Sleeping Beauty: if
Beauty isn’t certain that the procedure will go as promised (if she thinks maybe
the experimenters were lying about what they’d do), then awakening doesn’t
provide Beauty with evidence – for example – that the coin used was fair.
Automatic confirmation of the many-worlds hypothesis over the single-
world hypothesis for agents who are certain of GGG is altogether unsurprising,
and it need cause no alarm for Bayesian confirmation theorists. We have no
good reason to assign any significant credence to GGG, so it matters not at all
what our credences in the universe having branched conditional on GGG might
be. If we were somehow to come to believe that God tosses a coin to decide, for
every quantum interaction, whether the outcome was to be chancy or branchy,
then we would indeed have reason to favour the hypothesis that there are many
branches over the hypothesis that there is one branch. But this is all exactly as
it should be.
10. Non-chancy Sleeping Beauty cases
The foregoing arguments have an interesting upshot for variants of Sleeping
Beauty cases which lack the chancy element of the original case. In the
mathematical Sleeping Beauty case (MSB), uncertainty about the result of a fair
coin toss is replaced by uncertainty about the truth of a mathematical
proposition. On Sunday night Beauty has credence 1/2 that Fermat’s Last
Theorem is true. She will be awakened on Monday if the theorem is true, and on
both Monday and Tuesday (again with her memories from Monday erased) if the
theorem is false. Beauty knows all this. The puzzle is to say what credence
Beauty should have on Monday in the proposition that Fermat’s Last Theorem
is true (call this proposition True.)
24
The setup of MSB ensures that Beauty is not a fully rational Bayesian
agent, since such agents are required to be logically omniscient. Is there then
any sense in asking what Beauty’s credences ought to be? – after all, in one
obvious sense, her credence in True ought to be 1. I think this question does
make sense; it is common to distinguish diachronic and synchronic constraints on
credences, and to evaluate an agent’s performance with respect to the diachronic
constraints independently of her performance with respect to certain synchronic
constraints (such as logical omniscience). Epistemologists had certainly better
hope that something like this is possible: we’re not in fact perfect Bayesians, and
agents with non-trivial degrees of belief in mathematical propositions need to be
modelled by any plausible epistemology for mathematics.
The difference between SB and MSB may seem somewhat incidental.
General considerations relating to topic-neutrality perhaps make it natural to
assume that the two puzzles have the same solution. Regardless, discussions of
SB have usually paid little attention to the role that chance plays in the story18.
SB is usually described as a puzzle about self-location; and our two cases seem to
involve structurally similar self-locating uncertainty. In both SB and MSB, when
she awakens on Monday Beauty is unsure whether it is Monday or Tuesday.
In this section I will give some reasons for suspecting that the two puzzles
may in fact have different solutions. Two of the most powerful arguments for the
answer 1/3 in SB (both of which appear in Elga’s original article) are
inapplicable to MSB. Moreover, the analogy between SB and confirmation in
EQM gives us reason to prefer the answer 1/2 to MSB.
Consider first the long-run frequency argument for the answer 1/3. Beauty
knows that if SB is repeated infinitely many times, the ratio of Tails-awakenings
to Heads-awakenings will (with chance 1) tend to 2:1. Since any awakening is
indistinguishable from any other, her credence that a randomly-chosen
awakening is a Heads-awakening should be 1/3. This argument lapses in MSB.
The truth-value of mathematical propositions cannot vary between awakenings,
so Beauty knows that in the long-run either all the awakenings will be True-
awakenings or all the awakenings will be False-awakenings19.
18 For example, Christopher Meacham remarks that ‘the chanciness of the coin toss
only plays a superficial role in the argument.... the argument goes through just as
well if heads and tails are replaced by two different hypotheses we have other
reasons for having 1/2 / 1/2 credences in.’ (Meacham [2008]).
19 What if we pose different mathematical questions on each run of the experiment?
This variable-question mathematical case will probably pattern with SB rather than
with MSB, since it seems inevitable that the procedure by which the questions are
selected will involve a chance process at some stage or other. Be that as it may, in
this paper I will only address MSB.
25
Consider now the Principal Principle argument for the answer 1/3, set out
in detail in Section 6. This argument too lapses in MSB. Since mathematical
propositions have objective chances of either zero or one20, the constraint that
the Principal Principle places on credence in mathematical propositions amounts
to the requirement of logical omniscience, which is explicitly suspended in MSB.
Accordingly, nothing in the setup of MSB prohibits Beauty from adopting
credence 1/3 in True on being informed that it is Monday, and hence nothing
prohibits her from adopting credence 1/2 in True when she initially awakens on
Monday.
I conclude that there are important disanalogies between MSB and SB. Two
influential arguments for the thirder position in SB are inapplicable to the
mathematical case. This provides reason to doubt whether our two cases have a
uniform solution. Moreover, the analogy between MSB and confirmation in EQM
provides an indirect argument for the answer 1/2 in MSB.
11. Conclusions
The analogy between SB and Everettian confirmation breaks down in a
crucial way. Whether Heads or Tails is true depends on the outcome of a chance
process, while whether EQM or ST is true does not depend on the outcome of
any chance process. Consequently, SB involves the loss of relevant evidence –
evidence that the result of the coin is effectively chancy – while Everettian
confirmation scenarios involve no such loss. Although Bradley’s argument fails to
establish the halfer conclusion in the case of SB, it does succeed in dispelling the
spectre of automatic confirmation of EQM. Everettians need not adopt any novel
confirmation theory; given a correct conception of the evidence, EQM is
confirmed just in the same way as ordinary stochastic theories.
This result is doubly favourable to Everettians. If I am right, then Bayesian
confirmation theory can be combined with EQM in a straightforward and non-
pathological way, and Everettians are not obliged to adopt the unpopular halfer
position in SB. Opponents of EQM would do better to target its metaphysics
than to target its epistemology21.
20 This is so even according to ‘compatibilist’ conceptions of chance such as those of
Eagle [2011], Glynn [2010] and Handfield & Wilson [2014].
21 This paper was presented at two events at Monash University held in 2011 as
part of the AHRC ’Neglected Problems of Time’ project, led by Toby Handfield and
Graham Oppy. §10 was also presented at the Joint Session 2012. Thanks to these
audiences, and to an anonymous referee, for feedback; special thanks to Rachael
Briggs, Antony Eagle, Rohan French, Al Hájek, Toby Handfield, Jenann Ismael,
Peter Lewis, Chris Meacham, Dave Ripley and Wo Schwarz. My greatest debt is to
Miranda Rose, for the illustrations.
26
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